Library.

9 1963

^ecknlcai riote 92o. /*?

A COMPARISON OF TWO MELTING-PRESSURE EQUATIONS

CONSTRAINED TO THE TRIPLE POINT USING DATA FOR

ELEVEN GASES AND THREE METALS

ROBERT D. GOODWIN AND LLOYD A. WEBER

U. S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS
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NATIONAL BUREAU OF STANDARDS

^ecltnlcal <Vlote 1S3

Issued October 6, 1963

A COMPARISON OF TWO MELTING-PRESSURE EQUATIONS

CONSTRAINED TO THE TRIPLE POINT USING DATA FOR

ELEVEN GASES AND THREE METALS

Robert D. Goodwin and Lloyd A. Weber

Cryogenic Engineering Laboratory

National Bureau of Standards

Boulder, Colorado

NBS Technical Notes are designed to supplement the Bureau's regular publications program. They provide a means for making available scientific data that are of transient or limited interest. Technical Notes may be

listed or referred to in the open literature.

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25, D. C. Such permission is not needed, however, by the Government agency for which the Report hos been specifically prepared if that agency wishes to reproduce additional copies for its own use.
CONTENTS

1. Introduction 1

2. Data and Computational Method 3

3. Results 5

4. Summary and Conclusions 7

5. Acknowledgements 8

6. References 9

A COMPARISON OF TWO MELTING- PRESSURE EQUATIONS

CONSTRAINED TO THE TRIPLE POINT USING DATA FOR ELEVEN

GASES AND THREE METALS

Robert D. Goodwin and Lloyd A. Weber

Parameters have been determined by a least-squares

method for the reduced Simon equation and for a new,

empirical melting equation using data for H?

, Do, T?, Ne,

Ar, Kr, Xe, N , O , CO , H

2 C Na, K, and Hg. The new

equation, (P-P )/(

rT-T ) = Aexp(-a/T) + BT, represents

experimental data with essentially the same accuracy as

the Simon equation. It provides a sensitive difference

method for graphical examination of data.

1. Introduction i

Experimental data for melting curves often have been represented by the original Simon equation [Simon and Glatzel, 1929; Simon,

Ruhemann, and Edwards, 1930],

P = a + bT° (1-a)

with three parameters, omitting the triple-point datum. Since a number of triple -point determinations now are available, it is

appropriate to establish the two parameters of the reduced Simon

equation [ Simon, 1937; 1953],

P . p = p

t o (T/T

t )° - 1 (1)
as constrained to the triple point (subscript t_), using a uniform computational method based on relative, rather than absolute deviations

for all substances.

Whereas (1) has been derived with certain assumptions from

modern theory of solids [ Gilvarry, 1956; Glass, 1963; Salter, 1954;

Voronel, 1948, 1959], real or apparent difficulties are encountered

in the accurate empirical representation of some data, e. g. [ Michels

and Prins, 1962; Pistorius, Pistorius, Blakey, and Admiraal, 1963],

and for some metals the theoretically derived relation

c = (67 + D/(6y - 2)

(y = Grueisen's constant) is not confirmed experimentally [Strong and

Bundy, 1959] .

In view of these difficulties, we have at the same time examined

further the empirical equation recently used for interpolation near the

triple point of hydrogen [ Goodwin, 1962; Goodwin and Roder, 1963] ,

y = A exp(-a/T) + BT, (2)

where

y =(P- P

t )/(T - T

t

).

For this examination, substances other than low-boiling gases are

included for which, however, the triple -points may not be accurately

known. It is possible to deduce that (2) implies a temperaturedependence of the Grueisen constants, not found in the assumptions

of the latter equation of state. Gilvarry, however, has concluded that

a small temperature -dependence of these constants may be expected

[Gilvarry, 1956]. Figure 1 illustrates behavior of the experimental

function y with data for H , D , N , Ar, and CO . The small

temperature-dependence of this function (as compared with the pressure
P) provides a direct method for graphical examination of the relative

deviations or precision of a set of data, prior to the tedious iterative

determination of parameters either for (1) or (2).

Concerning helium, for which no triple -point exists, we merely

note, for example, that for-(a/T) « at very low temperatures, (2)

may be reduced to

P= P - (BT ) • T + BT 2

(2-a)

t t

and that this form is the same as used by Mills, Grilly and Sydor iak

(1961) near the melting press

0. 3 < T < 0. 5°K, namely,

3

(1961) near the melting pressure minimum for He in the range

P= 32.42- 21.25 T + 32. 20 T

2

.

Equation (1), on the other hand, clearly is monotonic.

2. Data and Computational Method

For each substance, the self-consistent data of but one investigator have been selected for present purposes. Original, unpublished

data on hydrogen, deuterium, tritium, neon, nitrogen, and oxygen

were generously given to us by Edward R. Grilly [Mills and Grilly,

1955; Mills and Grilly, 1956] . Argon, on the other hand, has been

selected for illustration using data of three investigators over a wide

range of pressures [ Bridgman, 1935; Lahr, and Eversole, 1962; and

Michels and Prins, 1962] . Further references to experimental data

are given by authors cited here. Triple-point constants used for the

present computations are given in table 1. All pressures are in

atmospheres [ Hilsenrath, 1955] . Absolute temperatures below 80°K

are on the NBS 1955 low-temperature scale [ Goodwin and Roder, 1963]
Since relative, rather than absolute, error is roughly constant

in the experimental determination of melting pressures, the leastsquares method (with trial variation of one parameter) has been

applied to constant or nearly constant forms of the above equations,

(P-P

t )/ (T/T

t

) C

- 1 = P , o

(P-P)/(T-T)T = AT"

1 e"a ' T

+ B.

(1-b)

(2-b)

In (2-b), all three constants have been treated as adjustable parameters, in contrast to the earlier treatment for hydrogen isotopes

[Goodwin, 1962] . Computational results include the root mean square (r.m. s. ) absolute deviation in atm, and the r. m. s. relative deviation in percent, defined respectively by

D =

A =

-1 1/2

(AP) /n

1 iA

1

• /»

where

S ^ 100(Pcalc- P)/ P

(3-a)

(3-b)

(3-c)

for a number n of datum points, and where A P is the difference

between calculated and experimental values. The form of (3-a) is the

same as used by Mills and Grilly [ 1956] . Minima in D and A,

however, generally do not occur for the same set of parameters.

Those corresponding to the minimum relative deviation, A, are

given in the following results, in contrast to those for a minimum

in D given by Mills and Grilly [ 1956] . To examine detailed behavior
near the triple point, the average of absolute values of deviations from

the first six data of each investigator at the lowest pressures are given

in percent in table 2 under the symbol § .

3. Results

Table 2 presents the sources of data and the number of points,

n, used in the least-squares determination of parameters, and the

maximum pressure of these data. There follow the parameters and

deviations for each equation. For hydrogen on lines 1 and 2, two

different sets of data are compared. Argon is examined with various

data. Line 6 is for the mutually consistent data of Michels and Prins

[ 1962] and of Bridgman [ 1935] . On line 7 (2) has been fitted to

synthetic data computed by (1), line 6. Line 8 includes, also, the

data of Lahr and Eversole to 18, 000 atm [ Lahr and Eversole, 1962] , whereas line 9 employs the latter data only.

A comparison of relative deviations, A, in table 2 shows in

general that (2) represents the data as well as, or, for some substances, somewhat better than (1). The improvement is clear for the

hydrogen data on line 1, for argon on line 6, for xenon, and for water.

Equation (2) contains a term which may be compared with the

internal pressure, P , of (1) at T = T.. The ratio of these terms

o t for the gases through O in table 2 is found to be

(AT /P ) exp (-a/T ) = 1. 2 ± 0. 1

with the exception of xenon for which the ratio is 1. 70.

For xenon, in table 3, the comparison of equations is given at

all points, since Michels and Prins found it necessary to fit the data

by (1-a), independently, in two regions [Michels and Prins, 1962] .
Whereas (2) gives a better average representation than (1), it does

not overcome the systematic deviation seen in this table.

Some of the results for argon are presented graphically. The

ordinate of the figure 2 logarithmic plot was computed with the internal pressure of Michels and Prins for (1-a), while the slope of the

extrapolated straight line corresponds to their values of c [Michels

and Prins, 1962] . Included for comparison are data of Bridgman

[1935], of Lahr and Ever sole [ 1962] , and of Robinson [ 1954] . As

compared to the pressure at 360°K (near 18, 000 atm. ), given by (1),

line 6 of table 2, the smoothed function of Lahr and Eversole yields a

value 7. 0% lower, and (2), line 6 of table 2, yields a value 3. 5%

higher, the latter equation being shown by the dashed curve of figure

1. Figure 3 for argon is derived from (2), using parameters from

line 6 of table 2. The straight line represents (2). The dashed line

represents (1). Datum points are the same as on figure 2. Figure

4 is similar, except that all constants are from line 8 of table 2.

This type of difference plot provides a highly sensitive method for

graphical examination of the precision of experimental data.

Table 4 for argon presents relative deviations of both equations

from the data of Bridgman [ 1935]; Lahr and Eversole [ 1962] ; and

Michels and Prins [1962] , line 8 of table 2 to 18, 000 atmospheres.

It is seen that the two equations give comparable representations of

these combined data to 18, 000 atm.

Table 5 for argon compares first derivatives, s = dP/dT, of

the two equations in terms of percent difference, 100(s - s )/s , in

which subscripts refer to the equations. Under heading (a) are given

results for the consistent data of line 6 of table 2, whereas, under (b),

the results correspond to the apparently less consistent data of line 8
of table 2. For each case, it is seen that the first derivatives of the

two equations are comparable over the range of the data.

Table 6 presents a comparison of the Simon constants recentlydetermined by S. E. Babb, Jr. [ 1963] , by a method of absolute

deviations, with those determined in the present report by a method

of relative deviations.

Since importance has been attached to the fact that (1) is a "reduced" form [Simon, 1937], the reduced constants for (2) are

examined in table 7. The following ranges of values are found for

the gases H through N? of table 2,

2 "

a/T ATP BT /F-

' t t t t' t 0.04—0.4 (3—17)- 10

3

(0.1—1.3) . 10

2

4. Summary and Conclusions

The two parameters of the reduced Simon equation have been

determined for a number of gases and some metals by a uniform

method which assumes that relative rather than absolute uncertainties

are roughly constant in the experimental pressures.

Some current empirical and theoretical difficulties with the

Simon equation, already mentioned, have led to examination of a new,

empirical equation with three parameters. The latter equation is

presented as a result of the remarkable equivalence to the Simon

equation found here in the empirical representation of data, even

though no simple mathematical equivalence between the equations is

apparent.
It may be concluded that (1) will be preferred to (2) for its

simplicity and qualitative theoretical support, except where the latter

gives improved empirical representation of data, or until such time

as it receives some theoretical interpretation. Equation (2), however,

provides a sensitive difference method for graphical examination of

the precision of data.

5. Acknowledgements

We are indebted to Roger M. Gallet of NBS for suggesting this

examination of the reduced form of the Simon equation and for pointing

out Lahr and Eversole, [ 1962] , and to John W. Stewart of The

University of Virginia for a relation between (2) and the Gruneisen

constants. Robert J. Corruccini of this laboratory kindly brought to

our attention Strong and Bundy, [ 1959] . The work has been

supported in part by the National Aeronautics and Space Administration.
6. References

[l] Babb, S. E. , Jr., (1963), Parameters in the Simon equation

relating pressure and melting temperature, Rev. Modern

Phys. 35, 400.

[Z] Babb, S. E. , Jr., (1963), Values of the Simon constants, Rev.

Modern Phys. 38, 2743.

[3] Bridgman, P. W. , (1914); (1926), Change of phase under

pressure, Phys. Rev. 3, 153, (1914); 27, 68 (1926).

[4] Bridgman, P. W. , (1935), The melting curves and compressibilities of nitrogen and argon, Proc. Am. Acad.

Arts. Sci. 70, 1.

[5] Danon, F., and K. S. Pitzer, (1962), Corresponding states

theory for argon and xenon, J. Phys. Chem. 66, 583.

[6] Gilvarry, J. J., (1956), Equation of the fusion curve, Phys.

Rev. 1_02_ (2), 325-31.

[7] Glass, Solomon J., (1963), The Gruneisen equation of state

Simon melting equation, Private communication.

[8] Goodwin, Robert D. , (Dec. 1962), Melting-pressure equation

for the hydrogens, Cryogenics 2, 353.

[9] Goodwin, R. D.,andH. M. Roder, (Mar. 1963), Pressuredensity-temperature relations of freezing liquid parahydrogen to 350 atm. , Cryogenics, 3(1) , 12.

[10] Grilly, E. R. , (1951), The vapor pressures of hydrogen,

deuterium and tritium up to three atmospheres, J. Am.

Chem. Soc. 73> 843.

[ll] Grilly, E. R. , (June, 1962), The vapor pressure of solid and

liquid neon, Cryogenics 2, 226.

[12] Hilsenrath, Joseph, editor, (1955), Tables of thermal properties of gases, NBS Circular 564.
[ 13] Hoge, Harold J. , (1950), Vapor pressure and fixed points of

oxygen and heat capacity in the critical region, J. Res. 44,

321.

[ 14] Lahr, P. H., and W. G. Eversole, (1962), Compression

isotherms of argon, krypton, and xenon through the

freezing zone, J. Chem. and Eng. Data, 7(1), 42.

[15] Michels, A., B. Blaisse,and J. Hoogschagen, (1942), The

melting curve of CO to 2800 atmospheres, Physica 9, 565.

[16] Michels, A., T. Wassenaar.and B. Blaisse, (1942), The

melting curve of Hg to 3000 atmospheres, Physica 9, 574.

[17] Michels, A., T. Wassenaar, Th. Sluyters.and W. DeGraaff,

(1957), The triple points of carbon dioxide and of argon as fixed points for the calibration of thermometers, Physica

_23, 89.

[18] Michels, A., and C. Prins (1962), The melting lines of argon,

krypton, and xenon up to 1500 atm; representation of the

results by a law of corresponding states, Physica 28, 101.

3

[19] Mills, R. L.,andE. R. Grilly, (1955), Melting curves of He , He4, H , D , Ne, N and O up to 3500 Kg/cm2

, Phys.

Rev. 9T, 48T).

[20] Mills, R. L., and E. R. Grilly, (1956), Melting curves of H,

D and T up to 3500 Kg/cm2

, Phys. Rev. 101, 1246.

2'

[21] Mills, R. L. , E. R. Grilly, and G. Sydoriak (1961), Anomalous

melting properties of He 3

, Annals of Physics 12, 41.

[22] Pistorius, C. W. F. T. , M. C. Pistorius, J. P. Blakey, and

L. J. Admiraal, (1963), Melting curve of ice VII to 200 k

bar, J. Chem. Phys. 38(3) , 600-602.

[23] Robinson, D. W. , (1954), An experimental determination of

the melting curves of argon and nitrogen into the 10, 000

atm. region, Proc. Roy. Soc. (London) A225, 393.

[24] Salter, L. , (1954), The Simon melting equation, Phil. Mag.

(7) 45, 369.

10
[ 25] Simon, Franz E., and G. Glatzel, (1929), Remarks on meltingpressure curves, Z. Anorg. Allgem. Chem. 1 78 , 309.

[26] Simon, F. E. , M. Ruhemann.and W. A. M. Edwards, (1930),

The melting curves of hydrogen, neon, nitrogen, and argon,

Z. Phys. Chem. B6 , 331.

[ 27] Simon, F. , (1937), On the range of stability of the fluid state,

Trans. Faraday Soc. 33 , 65.

[ 28] Simon, F. , (1953), The melting of iron at high pressures,

Nature 172 , 746.

[29] Strong, H. M. , and F. P. Bundy, (1959), Fusion curves of

four group VIII metals to 100, 000 atmospheres, Phys.

Rev. 115, 278.

[30] Voronel', A. V., (1948); (1959), On the equation of the melting

curve, Zh. tekh. Fiz. _28(11), 2630-4(1948), in Russian;

Physics Abstracts 62, 10941 (1959).

[31] Woolley, H. W. , R. B. Scott, and F. G. Brickwedde, (1948),

Compilation of thermal properties of hydrogen in its

various isotopic and ortho-para modifications, J. Res. 41,

379.

11
Table 1. Triple -point constants used.

Substance T

t

, °K

13. 947

P , atm. Reference

n-H

2

0. 071 [31]

n~D

2

18. 72 0.169 : io]

n-T

2

20.61 0.213 :io]

Ne 24. 544 0.427 :n]

Ar 83.812 0.685 :i7, is]

Kr 115.745 0.724 ! :is]

Xe 161. 364 0.806 | .18]

N

2

63.146 0.124 | :iz]

°2 54. 353 0.0015 | .13]

C°2 216. 577 5. 11 | .17, 12]

H

2

354. 7 5*

f 21,260.* | 22]

Na 370. 75| 0.9684t 1 3]

K 335. 65t 0.9684t 1 3]

Hg 234. 32| 0. [ 17]

* Triple -point ice VI + ice VII + liquid.

| The melting-point of original data.

12

I

IE
Table 2. Constants and deviations for the two equations

Line

Substance

Data

P

max

Equation 1 Equation i >

Reference n c P

o D 6 A a A B D 6 A

1 n-H

2

[31] 15 608 1.930 216. 72 2. 7 0. 52 0. 75 6.2 32. 859 . 60768 0. 8 0. 50 0.43

2 n-H

2

[20] 26 3, 600 1. 795 244. 89 15. 7 2. 06 1.41 4. 1 28. 803 . 67444 10.4 2.43 1.45

3 n-D

2

[20] 23 3, 500 1.815 406. 07 7. 7 0. 56 0.49 4.4 33, 055 . 67890 4.6 0. 65 0.44

4 n-T

2

[20] 15 3, 000 1. 780 509.88 3. 6 1. 32 1. 14 1. 30.869 . 72807 6.5 1. 34 1.16

5 Ne [19] 36 3,400 1. 630 985. 06 6.2 0.47 0.46 4. 59. 334 . 59879 5. 7 0.46 0.47

6. Ar [4,18] 16 5,800 1. 594 2,087. 7 3. 3 0. 03 0. 09 7. 2 33. 275 . 10971 1. 0. 03 0.05

7. Ar 16 5,800 9.0 34. 294 . 10596 1. 32 0. 03 0. 04

8 Ar [4,14,18] 30 18, 000 1. 521 2, 213. 3 89.9 0.75 1.47 24. 45. 413 . 06264 75. 0. 39 1. 33

9 Ar [14] 14 18, 000 1. 510 2, 242. 2 108. 1 1. 95 1. 80 48. 56. 334 . 04234 119. 2 1.66 1. 57

10 Kr [18] 14 1, 500 1. 615 2,349. 0. 1 0. 12 0. 11 5. 24. 949 . 07680 0. 2 0. 12 0.11

11 Xe [18] 19 1, 500 1.710 2, 352. 3 2. 1 0.46 0. 42 71. 38. 580 0. 1. 0. 29 0. 26

12 N

3 [19] 24 3, 500 1. 790 1, 586. 2 2. 1 0.30 0. 22 7. 33. 133 . 24087 2. 0. 31 0. 21

13

°Z [19] 16 3, 500 1. 755 2, 666.8 5. 1 0. 60 0. 47 14. 83. 374 . 38743 5.4 0. 59 0.44

14 C0

2 [15] 25 2,800 3. 1 3,178.3 5.4 2.48 1.65 0. - 7. 05304 . 242083 4.9 2. 55 1. 64

15 H

2 [22] 26 196,000 4. 8 6, 048. 3 4370. 2. 30 5. 13 450. -5546. 02 4.80652 3005. 2. 30 3. 58

16 Na [S] 12 11, 600 3.6 11,409. 22. 1 0. 76 0. 74 380. -1530. 54 1. 78578 26. 2 0. 74 0. 62

17 K [3] 12 11, 600 4.4 4, 224.

4

31.3 0. 37 0.44 280. - 503.249 . 815969 39.4 0. 35 0.49

18 Hg [16] 13 3,000 0. 76 59,249. 3.6

13

0.99 0. 94 260. -8872.22 13. 3083 2. 4 0.94 0. 88

Table 3. Comparison of (1) and (2) for Xenon

T P

5. 53

6

i 6

2

161. 554 0. 33 0. 00

162.439 27.49 0.66 0. 39

163. 375 50.82 0. 67 0.45

164. 171 70.95 0. 37 0. 19

167. 154 147. 31 -0. 23 -0. 28

171.455 259. 23 -0. 51 -0. 42

176.978 405.89 -0. 64 -0.42

184. 004 596. 25 -0. 56 -0. 26

191.144 794. 08 -0. 39 -0. 09

197.868 984.49 -0. 22 0. 02

203. 000 1132. 20 -0. 06 0. 09

207. 205 1255.18 0. 05 0. 10

211. 142 1371. 51 0. 17 0. 11

215. 264 1494. 53 0. 30 0. 11

0. 37 0. 21

15
rg to o CO vD rg O CO r^ CO vD c- 1—1 o LD CO o

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1—1 o SO (M "* oo CO i—

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16
Table 5. Comparisons of s = dP/dT for Argon

T

100 <s 2

- sjj/s,

(a) (b)

83. 812 +. 06 -2. 02

83. 900 +. 07 -2. 00

84. 000 +. 07 -1. 97

84. 500 +. 06 -1. 83

85. 000 +. 06 -1. 70

86 +. 06 -1.44

87 +. 06 -1. 21

88 +. 05 - .98

90 +. 04 - . 58

100 -.06 + . 80

110 -. 15 +1.42

120 -. 21 +1. 61

140 -.13 +1. 34

160 +. 18 +0. 74

180 +. 69 +0. 10

200 +1. 35 -0.47

240 -1. 30

280 -1. 71

320 -1.76

360 -1. 54

(a) Constants from line 6 of table 2.

(b) Constants from line 8 of table 2.

17
Table 6. Comparison of Simon Constants

of S. E. Babb, Jr. [1963] with those of table 2.

Substance ^o^Babb'^o* table 2

1. 1051

* C) Babb/C Uable 2

n-H

2

0.9716

n-D

2

1. 0443 0. 9847

3 n " T

2

1. 0258 0.9911

( Ne 1. 0395 0.9815

Av 0. 9992 0.9994

Kr 0.9983 1. 0012

C Xe 1. 0951 0.9294

( N

2

0.9996 1. 0006

i °2 1. 0114 0. 9929

j co

2* 1. 2421 0. 8387

1 HzO* 2. 1180 0. 6479

Na 1. 0355 0. 9814

J K 0. 9975 1. 0091

1 Hg* 0. 6366 1. 5487

* Different data used by respective authors,

18
Line *

Table 7. Approximate reduced constants for equation (2)

Substance a/T (AT /P )

• 10" 3

(BT /P )

• 10~'

1 H 0.445 6.45 1.194

2 H

2

0.294 5.66 1.325

3 D 0.235 3.66 0.752

4 T 0.049 2.99 0. 704

5 Ne 0.163 3.41 0.344

6 Ar 0.086 4.07 0.134

7 Ar 0.107 4.20 0.130

8 Ar 0.286 5.56 0.076

9 Ar 0.573 6.89 0.052

10 Kr 0.043 3.89 0.120

11 Xe 0.440 7.72

12 1M 0.111 16.87 1.227

13 O 0.258 3021. 140.

14 CO

z -0.30 0.103

15 H

z O 1.27 -0.09 .0008

16 Na 1.03 -586. 6.84

17 K 0.83 -175. 2.83

Lines correspond to table 2.

19
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GPO 836-475

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Luther H. Ho>l KEAUOF STANDARDS

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